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An orthogonal projection is a projection for which the range ... This formula can be generalized to orthogonal projections on a subspace of arbitrary dimension.
Conversely, every projection operator of rank defines a subspace := as its image. Since the rank of an orthogonal projection operator equals its trace , we can identify the Grassmann manifold G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} with the set of rank k {\displaystyle k} orthogonal projection operators P {\displaystyle P} :
In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace of a vector space equipped with a bilinear form is the set of all vectors in that are orthogonal to every vector in .
In the Hilbert space view, this is the orthogonal projection of onto the kernel of the expectation operator, which a continuous linear functional on the Hilbert space (in fact, the inner product with the constant random variable 1), and so this kernel is a closed subspace.
Hilbert projection theorem — For every vector in a Hilbert space and every nonempty closed convex , there exists a unique vector for which ‖ ‖ is equal to := ‖ ‖.. If the closed subset is also a vector subspace of then this minimizer is the unique element in such that is orthogonal to .
The Gram–Schmidt process takes a finite, linearly independent set of vectors = {, …,} for k ≤ n and generates an orthogonal set ′ = {, …,} that spans the same -dimensional subspace of as . The method is named after Jørgen Pedersen Gram and Erhard Schmidt , but Pierre-Simon Laplace had been familiar with it before Gram and Schmidt. [ 1 ]
A popular extension of Matching Pursuit (MP) is its orthogonal version: Orthogonal Matching Pursuit [14] [15] (OMP). The main difference from MP is that after every step, all the coefficients extracted so far are updated, by computing the orthogonal projection of the signal onto the subspace spanned by the set of atoms selected so far. This can ...
The vector projection (also known as the vector component or vector resolution) of a vector a on (or onto) a nonzero vector b is the orthogonal projection of a onto a straight line parallel to b. The projection of a onto b is often written as proj b a {\displaystyle \operatorname {proj} _{\mathbf {b} }\mathbf {a} } or a ∥ b .